Linear recurrences with constant coefficients: the multivariate case

While in the univariate case solutions of linear recurrences with constant coe cients have rational generating functions we show that the multivariate case is much richer even though initial conditions have rational generating functions the corresponding solutions can have generating functions which are algebraic but not rational D nite but not algebraic and even non D nite Introduction The aim of this paper is to study the nature of multivariate generating functions F x xd X n nd an ndx n xd d X